Additive Chow groups with higher modulus and the generalized de Rham-Witt complex [Elektronische Ressource] / von Kay Rülling
89
pages
English
Documents
2005
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe en t'inscrivant gratuitement
Découvre YouScribe en t'inscrivant gratuitement
89
pages
English
Documents
2005
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Publié par
Publié le
01 janvier 2005
Nombre de lectures
34
Langue
English
Additive Chow groups with higher modulusandthe generalized de Rham-Witt complexDissertationzur Erlangung des GradesDoktor der Naturwissenschaften(Dr. rer. nat.)vorgelegt beimFachbereich Mathematik und Informatikder Universit¨at Duisburg-Essenvon Kay Rull¨ ingaus EssenTag der Disputation 11. Mai 2005Vorsitzender der Pruf¨ ungskommission Prof. Dr. Axel KlawonnErster Gutachter Prof. Dr. H´el`ene EsnaultZweiter Gutachter Prof. Dr. Eckart ViehwegAcknowledgmentsI am very grateful to H´el`ene Esnault for giving me the subject of this thesis and forher excellent guidance and support, not only during the work on this problem, butfrom the very beginning of my mathematical studies.IwanttoexpressmygratitudetoLarsHesselholtforexplainingtheWittvectorsand the generalized de Rham-Witt complex to me.I would like to thank Spencer Bloch and Gerd Faltings for several hints, inparticular, to construct the trace for the Witt vectors via the norm map.I want to thank Stefan Kukulies for the many discussions concerning the foun-dations of algebraic geometry, Andre Chatzistamatiou for the careful reading of theintroduction and Wioletta Syzdek for the supply with chocolate.Finally I want to thank my parents for making untroubled studies possible.
Voir
Publié par
Publié le
01 janvier 2005
Nombre de lectures
34
Langue
English
Additive
the
Chow
gro
generalized
ups with higher modulus and e Rham-Witt complex
d
Dissertation zur Erlangung des Grades Doktor der Naturwissenschaften (Dr. rer. nat.)
vorgelegt beim Fachbereich Mathematik und Informatik der Universit¨t Duisburg-Essen a
Tag der Disputation
vonKayRu¨lling aus Essen
VorsitzenderderPr¨ufungskommission Erster Gutachter Zweiter Gutachter
11. Mai 2005
Prof. Dr. Axel Klawonn Prof.Dr.He´le`neEsnault Prof. Dr. Eckart Viehweg
Acknowledgments
IamverygratefultoHe´le`neEsnaultforgivingmethesubjectofthisthesisandfor her excellent guidance and support, not only during the work on this problem, but from the very beginning of my mathematical studies. I want to express my gratitude to Lars Hesselholt for explaining the Witt vectors and the generalized de Rham-Witt complex to me. I would like to thank Spencer Bloch and Gerd Faltings for several hints, in particular, to construct the trace for the Witt vectors via the norm map. I want to thank Stefan Kukulies for the many discussions concerning the foun-dations of algebraic geometry, Andre Chatzistamatiou for the careful reading of the introduction and Wioletta Syzdek for the supply with chocolate. Finally I want to thank my parents for making untroubled studies possible.
Contents
0
1
2
3
4
A
Introduction
Witt Vectors
The generalized de Rham-Witt Complex
A Residue Theorem
Additive cubical Chow groups with higher modulus
Intersection Theory for Cartier Divisors
References
1
5
14
29
54
77
81
0
Introduction
1
LetXbe an equidimensional scheme of finite type over a fieldkand write Δn= Speck[t0 . . . tn]/(Pni=0ti−1). In [Bl86] Bloch introduced higher Chow groups CHp(X n), generalizing the Chow groups ofX(i.e. CHp(X0) = CHp(X)). Roughly speaking, these are defined by considering the quotient ofp-codimensional cycles in X×Δnin suitable good position modulo the boundary ofp-codimensional cycles inX×Δn+1, where the boundary is given by intersecting with the faces (ti= 0) and then take the alternating sum. This construction is usually referred to as the simplicial definition of the higher Chow groups. There is also a cubical one, which mainly differs by taking (P1\ {1})ninstead of Δn(see [To92]) and one can show that these two definitions of Bloch’s higher Chow groups coincide. One of the rare cases, where the higher Chow groups can be computed, which means to give a presentation in terms of generators and relations, is the caseX= Speckandp=n, i.e. formal sums of points in Δn(resp. (P1\{01∞})n) modulo the boundary of formal sums of curves in good position in Δn+1(resp. (P1\ {1})n+1 this case Nesterenko-Suslin). In and Totaro proved (see [NeSu89], [To92])
CHn(k n)∼=KnM(k)
whereKnM(k) are the degreenelements in the Milnor ring ofk. We observe that one could replace Δnin the definition of the higher Chow groups by Speck[t0 . . . tn]/(Pti−λ), for anyλ∈k×. In [BlEs03a] Bloch and Esnault investigated the degenerated caseλ They obtain a theory of additive higher= 0. Chow groups, SHp(X n), and prove in particular
−1 SHn(k n) =∼Ωk/nZ
using a presentation of Ωn/k−Z1as a quotient of the anti commutative graded ring k⊗ZV∗k×modulo the graded ideal generated bya⊗a+ (1−a)⊗(1−a). In [BlEs03b, Section 6.] Bloch and Esnault construct a cubical version of the additive higher Chow groups (so far only for fields and on the level of zero cycles). Since these groups are our main object of study, we give a more precise definition. Denote by Zd(X) the group ofd-dimensional cycles onXand write
Xn=Gm×(P1\ {1})n
with coordinates (x y1 . . . yn).
Define Z1(Xn; 1) to be the subgroup of Z1(Xn)which is freely generated by 1-dimensional subvarietiesC⊂Xn,C6⊂Si(yi= 0∞) satisfying the following prop-erties (a) (Good position) (yi=j).[C]∈Z0(Xn−1\Si(yi= 0∞)), fori= 1 . . . n, j= 0∞.
(b)
(Modulus2condition) Ifν compactification ofC, then
(1)
:
C→P1×P1n e
n 2[ν∗(x= 0)]≤X[ν∗(yi i=1
= 1)]
is the normalization
e in Z0(C).
of
the
2
There is a map∂=Pni=1(−1)i(∂i0−∂i∞) Bloch and Esnault define
0
INTRODUCTION
: Z1(Xn; 1)→Z0(Xn−1\Yn−1) and
THn(k n) = THn(k n; 1) Z0(Xn−1\Yn−1) = ∂Z1(Xn; 1).
They show by similar methods as before, assuming 1/6∈k,
n−1 THn(k n =; 1)∼Ωk/Z.
Now the natural question is, what happens, if we replace the modulus 2 condition in (b), by a modulus (m condition, i.e. replace the 2 in (1) by (+ 1)m+ 1) and THn(k n; 1) by THn(k n;m)? The answer to this question, which is given in this thesis, is motivated by the following considerations. Denote by Pic(Ak1(m+ 1){0}) the equivalence classes of divisors inA1, supported onGm divisors. TwoD,D0being equivalent iffD−D0= div(f /h), for somef h∈1 +tk[t] withf−h∈tm+1k[t], this isf /h≡1 mod (m+ 1){0}in the language of [Se88, Chapter III,§1]. Now given such functionsfandhwe can define a curveC⊂Gm×P1\ {1}by the equation h(x)y−f(x) = 0 and one easily checks that this curve satisfies the modulus (m+ 1)-condition. Thus we obtain a well defined and surjective map
(2)
Pic(A1(m+ 1){0})−→TH1(k1;m).
Now it is quiet reasonable to believe this map to be an isomorphism (at least for m= 1, both sides equalk we may identify Pic(, by the above). ButA1(m+ 1){0}) with the group+11t+mt+k[1kt[]t]×, which in turn may be identified with the ring of generalized Witt vectors of lengthmoverk,Wm(k we hope to Hence), (see [Bl78]). obtain an isomorphism TH1(k1;m)∼=Wm(k). If this is true, then the groups THn(k n;m) should in general give something, which generalizestheabsoluteK¨ahlerdifferentialsontheonehandandthebigWittrings of finite length on the other. The natural suspect for this is the generalized de Rham-Witt complex of Hesselholt-Madsen,WmΩkn, which generalizes thep-typical de Rham-Witt complex of Bloch-Deligne-Illusie. And indeed the main theorem of this thesis is
Theorem.Letkbe a field of characteristic6= 2 we have for all. Thenn m≥1an isomorphism THn(k n;m) =∼WmΩnk−1.
The first part of the proof, namely to define the map from the additive Chow groups to the de Rham-Witt complex, is analogous to the proof of the casem= 1 by Bloch and Esnault. To verify that the map THn(k n;m)→WmΩkn−1is well defined, we use a reciprocity law (as it was done in [NeSu89], [To92], [BlEs03a] and [BlEs03b]). Here it is a ”sum-of-residues-equal-zero” theorem, which we prove in section 3, generalizing the well known residue formula for differentials on smooth projective curves. To obtain the inverse map we use the universality of the de Rham-Witt complex, i.e. we equip the additive Chow groups with a structure of
